quantitative analysis exam review solutions
MBAB 5P02 FINAL EXAM PREPARATION
#3. INVESTMENT STRATEGY
1.
The first step is to develop a linear programming model for maximizing return subject to constraints for funds available, diversity, and risk tolerance.
Let G = Amount invested in growth fund
I = Amount invested in income fund
M = Amount invested in money market fund
MAX 0.18G + 0.125I + 0.075M
S.T.
1)
2)
3)
4)
5)
6)
7)
G + I + M ≤ 800000
0.8G -0.2I -0.2M ≥ 0
0.6G -0.4I -0.4M ≤ 0
-0.2G +0.8I -0.2M ≥ 0
-0.5G +0.5I -0.5M ≤ 0
-0.3G -0.3I +0.7M ≥ 0
0.05G + 0.02I -0.04M ≤ 0
G ≥ 0; I ≥ 0; M ≥ 0;
Funds Available
Min growth fund
Max growth fund
Min income fund
Max income fund
Min money market fund
Max risk
The optimal objective function value is 94,133.336.
#4. TOYZ
Let
X = the amount (in $) of a six-month loan (at 11% interest rate) used by Toyz
Yi = the amount (in $) of monthly loan (at 5% interest per month) used by Toyz at month i;
Si = the amount (in $) of monthly saving by Toyz for month i;
(i=J,A,S,O,N,D)
Min
Z = 0.11X + 0.05 (YJ + YA + YS + YO + YN + YD)
s.t.
($ amount of money available at month i = $ amount of money spent at month i)
X + YJ + 20,000 = SJ + 60,000
SJ + YA + 30,000 = SA + 60,000 + 1.05YJ
SA + YS + 40,000 = SS + 80,000 + 1.05YA
SS + YO + 50,000 = SO + 30,000 + 1.05YS
SO + YN + 80,000 = SN + 30,000 + 1.05YO
SN + YD + 100,000 = SD + 20,000 + 1.05YN
1.11X + 1.05YD
≤
SD
X ≥ 0; Yi ≥ 0; Si ≥ 0 for all i=J,A,S,O,N,D
#5. HANSEN CONTROLS: TRANSPORTATION PROBLEM
Let
Xij = the number of panels shipped from source i to destination j (i=S,H,T,L,P; j=S,D,K)
Yi = 1 if plant i is built; = 0 otherwise (i = T,L,P)
Min
Z = 350000YT + 200000YL + 480000YP + 5XSS + 7XSD + 8XSK + 10XHS + 8XHD + 6XHK
+ 9XTS + 4XTD + 3XTK + 12XLS + 6XLD + 2XLK + 4XPS + 10XPD + 11XPK
s.t.
XSS + XSD + XSK ≤ 2500
XHS + XHD + XHK ≤ 2500
XTS + XTD + XTK ≤ 10000YT
XLS + XLD + XLK ≤ 10000YL
XPS + XPD + XPK ≤ 10000YP
XSS + XHS + XTS
#3. INVESTMENT STRATEGY
1.
The first step is to develop a linear programming model for maximizing return subject to constraints for funds available, diversity, and risk tolerance.
Let G = Amount invested in growth fund
I = Amount invested in income fund
M = Amount invested in money market fund
MAX 0.18G + 0.125I + 0.075M
S.T.
1)
2)
3)
4)
5)
6)
7)
G + I + M ≤ 800000
0.8G -0.2I -0.2M ≥ 0
0.6G -0.4I -0.4M ≤ 0
-0.2G +0.8I -0.2M ≥ 0
-0.5G +0.5I -0.5M ≤ 0
-0.3G -0.3I +0.7M ≥ 0
0.05G + 0.02I -0.04M ≤ 0
G ≥ 0; I ≥ 0; M ≥ 0;
Funds Available
Min growth fund
Max growth fund
Min income fund
Max income fund
Min money market fund
Max risk
The optimal objective function value is 94,133.336.
#4. TOYZ
Let
X = the amount (in $) of a six-month loan (at 11% interest rate) used by Toyz
Yi = the amount (in $) of monthly loan (at 5% interest per month) used by Toyz at month i;
Si = the amount (in $) of monthly saving by Toyz for month i;
(i=J,A,S,O,N,D)
Min
Z = 0.11X + 0.05 (YJ + YA + YS + YO + YN + YD)
s.t.
($ amount of money available at month i = $ amount of money spent at month i)
X + YJ + 20,000 = SJ + 60,000
SJ + YA + 30,000 = SA + 60,000 + 1.05YJ
SA + YS + 40,000 = SS + 80,000 + 1.05YA
SS + YO + 50,000 = SO + 30,000 + 1.05YS
SO + YN + 80,000 = SN + 30,000 + 1.05YO
SN + YD + 100,000 = SD + 20,000 + 1.05YN
1.11X + 1.05YD
≤
SD
X ≥ 0; Yi ≥ 0; Si ≥ 0 for all i=J,A,S,O,N,D
#5. HANSEN CONTROLS: TRANSPORTATION PROBLEM
Let
Xij = the number of panels shipped from source i to destination j (i=S,H,T,L,P; j=S,D,K)
Yi = 1 if plant i is built; = 0 otherwise (i = T,L,P)
Min
Z = 350000YT + 200000YL + 480000YP + 5XSS + 7XSD + 8XSK + 10XHS + 8XHD + 6XHK
+ 9XTS + 4XTD + 3XTK + 12XLS + 6XLD + 2XLK + 4XPS + 10XPD + 11XPK
s.t.
XSS + XSD + XSK ≤ 2500
XHS + XHD + XHK ≤ 2500
XTS + XTD + XTK ≤ 10000YT
XLS + XLD + XLK ≤ 10000YL
XPS + XPD + XPK ≤ 10000YP
XSS + XHS + XTS